The angle $\theta_1$ is located in Quadrant $\text{I}$, and $\sin(\theta_1)=\dfrac{11}{61}$. What is the value of $\cos(\theta_1)$ ? Express your answer exactly. $\cos(\theta_1)=$
Explanation: We have the value of $\sin(\theta_1)$ and we want to find the value of $\cos(\theta_1)$. We can do this using the Pythagorean identity: $\sin^2(\theta_1)+\cos^2(\theta_1)=1$ [How did we get the Pythagorean identity?] Solving for $\cos(\theta_1)$, we get the following: $\cos(\theta_1)=\pm\sqrt{1-\sin^2(\theta_1)}$ Plugging $\sin(\theta_1)=\dfrac{11}{61}$, we get the following: $\begin{aligned} \cos(\theta_1)&=\pm\sqrt{\dfrac{3600}{3721}} \\\\ &=\pm\dfrac{60}{61} \end{aligned}$ To figure out the sign of $\cos(\theta_1)$, we should use the fact that $\theta_1$ is located in Quadrant $\text{I}$. Since $\theta_1$ is in Quadrant $\text{I}$, $\cos(\theta_1)$ is positive. Therefore, this is the answer: $\cos(\theta_1)=\dfrac{60}{61}$